Set \(g=\mathrm {sign}(f)|f|^4\) , so that \(\int _Y \mathrm {sign}(g)|g|^{1/2}d\mathrm {vol}=0\) , hence
\(\Vert g\Vert _{L^{3/2}}=\inf _{a\in \mathbb {R}}\Vert g-a\Vert _{L^{3/2}}\) 
(see the discussion in [1]}). By definition of the Sobolev constant we hence obtain
\(\mathfrak {S}_Y\Vert f\Vert ^{12}_{L^6}&=\mathfrak {S}_Y\Vert g\Vert ^3_{L^{3/2}}\le \Vert dg\Vert ^3_{L^1}\\&=\Vert 4f^3df\Vert ^3_{L^1}\le 64\Vert f\Vert _{L^6}^{9}\Vert df\Vert ^3_{L^2}.\) 

where the last inequality follows from Cauchy-Schwartz. Hence
\(\mathfrak {S}_Y\Vert f\Vert ^{3}_{L^6}\le 64 \Vert df\Vert ^3_{L^2}\) 
and the result follows.
